Quick update..... The starter now engages the flywheel without problem. I guess that a half charged battery just couldn't spin it fast enough to push the bendix drive out. It took some spinning cycles and maybe a love tap with a hammer or two, but it's not engaging every single time.
Which is good, but.... Small problem, probably related to the ignition switch not spinning the starter: there is a wire in that bundle that gets HOT when the key is on. It's the pink one. Funny thing is, Toyota doesn't even mention the possibility of a pink wire in the wiring diagram, so I'm at a loss as to what it is. It looks like the pink wire runs out to a resistor or something, where it then terminates, and runs out of the other side as a B-Y wire (which is on whenever the key is on).
Funny thing is, I just disconnected all of the gas engine smog stuff, so if it's getting this hot now, I must have a short inside the wiring bundle or something. I doubt I'll have time to troubleshoot it tomorrow before I head back to CO, but I'll probably try...
Dan